Abstract:In the present paper we present the tensor-product approximation of
multi-dimensional convolution transform discretized via
collocation-projection
scheme on the uniform or composite refined grids.
Examples of convolving kernels are given by the classical Newton,
Slater (exponential) and Yukawa potentials,
, and
with .
For piecewise constant elements on the uniform grid of size ,
we prove the quadratic convergence in the mesh parameter
h=1/n, and then justify the Richardson
extrapolation method on a sequence of grids that improves the order of
approximation up to .
The fast algorithm of complexity is described
for tensor-product convolution on the uniform/composite grids of size ,
where are tensor ranks of convolving functions.
We also present the tensor-product convolution scheme in the two-level
Tucker-canonical format and discuss the consequent rank reduction strategy.
Finally, we give numerical illustrations confirming:
(a) the approximation
theory for convolution schemes of order and ;
(b) linear-logarithmic scaling
of 1D discrete convolution on composite grids;
(c) linear-logarithmic scaling in n of our tensor-product
convolution method on grid in the
range .